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LinearLinear Transformation Transformation with CG & animationwith CG & animation

Ogose Ogose ShigekiShigekiKawaiKawai--JukuJuku

Tokyo, JapanTokyo, Japanhttp://mixedmoss.com/atcm/2012/http://mixedmoss.com/atcm/2012/

1. Advantages of using Computer Graphics (CG).

• Grids can be drawn easily.

• Effects of changing the ‘original objects’ or ‘matrix’ can be seen immediately.

• Exotic objects such as photos can be transformed.

• Animations can be used.

2

2 1 Transformatioex n by

1 11.

−

original transformed

muffine.jar

(2,1)(-1,1)

Example of linear

OA 3OB OA' 3O2

i

B'2

ty

+ → +

(1,0)

(0,1)

Transformation of a photo bcos sin

ex2 )sin c

y s

. (o

Rθ θ

θθ θ

−=

original transformed

muffine.jar

Use a palette or type it in the field.

cos30 ,sin( )30° °( ,sin 30 c )os30° °−

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2 1 Animation of Rotation&Enlargement by

1 2ex3. F

− =

animation by rotation

current matrix is1 0

0 1

current matrix is2 1

1 2

−

is right-at-the-moment matrix. Which starts from the unit matrix and finishes as the target matrix.Curr matrixent

2. EigenVectors & Animation

Animation is useful for rotation - which has no real eigenvectors- , but it works even better for transformationswhich have real eigenvectors.

Next example has 2 eigenvectors.

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1 1 5 ,

0 02

10 10A A

=

=

5 0

For 0 2

A =

，

1 0eigenvectors : & eigenvalues :5&2 , respectively.

0 1

，

3 2For

1 4B =

，2 2

· 21 1

5 ,1 1 11

·BB− −

=

=

1 2eigenvectors : & eigenvalues :5&2 , respectively.

1 1−

，

Comparison of 2 transformations which have same eigenvalues.ex4.

5 0Transformation by . When the base is

1 0 an

0 02d

1A =

original transformed

animation by eigenvectors

(5,0)

(0,2)

muffine.jar

5

3 21 4

Transformation by . When the base is 1 0

and 0 1

B

=

When the base is (1,0) , transformation looks one and (0,1 of m) any.

(3,1)

(2,4)

1 0base is &

0 1

muffine.jar

1 2

3 21 4

Transformation by . When the 1 2

base is and 1 1

eB e−

= =

=

When the base is eigenvectors, transformation is easier to understand.

(1,1)(-2,1)

(5,5)

(-4,2)

1 2base is &

1 1−

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and have the same eigenvalues, thus they work similarly.Eigenvectors&values decide how linear transformations work.A B

by A

by B

by Bby Aby B

by Bby Aby A

Choose the base here.e1&e2 are the base set by you. (left)

1

2

1

1

shows the similarity between A&B. represents

the same transformation as , but it's the expression by and .

P BP P BP

B e e

− −

Choose (1/P)AP